liquidity flows and fragility of business...

LIQUIDITY FLOWS AND FRAGILITY OF BUSINESS ENTERPRISES

Wouter J. den Haan, Garey Ramey and Joel Watson∗

July 1998

Revised June 2002

Abstract. This paper develops a macroeconomic model in which funds are

channeled to entrepreneurs through long-term relationships between entrepreneurs and

lenders. The flow of funds to a relationship may be low, causing it to break up due to

insufficient liquidity. Multiple Pareto ranked steady states emerge from complemen-

tarity between financial intermediation, reflected by the number of relationships, and

households’ incentives to provide funds. This complementarity also serves as a mech-

anism for propagating aggregate shocks. Financial collapse may become inescapable if

a shock destroys sufficiently many relationships.

1. Introduction

The standard credit-market paradigm in macroeconomics presumes that firms borrow on

frictionless spot markets from anonymous lenders. An important body of evidence has

shown, however, that credit-market trading often takes place within long-term relationships

between borrowers and lenders; further, these relationships operate with frictions.1 In this

paper, we demonstrate the important role that such frictions play in sustaining low-activity

steady states and propagating aggregate shocks.

∗ den Haan: University of California, San Diego, CEPR and NBER. Ramey and Watson: University

of California, San Diego. We thank Andy Atkeson, Olivier Blanchard, Ricardo Caballero, Dean Corbae,

Jason Cummins, Mark Gertler, John McMillan, Valerie Ramey, Richard Rogerson, Chris Woodruff, Michael

Woodford, an anonymous referee, and numerous seminar participants for their comments. The authors

thank the NSF for financial support under grants SBR-965868 and SES-9975277.

1Section 2.4 provides an overview of this evidence.

1

LIQUIDITY FLOWS AND FRAGILITY OF BUSINESS ENTERPRISES 2

We develop a dynamic equilibrium model in which funds are channeled to entrepreneurs

through long-term relationships with lenders. Our model identifies three frictions that affect

this flow. First, there is a matching friction in the market to establish entrepreneur-lender

relationships. This friction highlights the importance of long-term relationships, because an

entrepreneur who is separated from his current lender must engage in time-consuming search

to find another source of funds.

Second, our model specifies an allocation friction in the distribution of funds to lenders.

We assume that random shocks affect the amount of funds available in a relationship, and

that it is costly for lenders to obtain additional funds on short notice.2 This friction can

cause the ex post allocation of funds to be suboptimal.

Third, our model posits contractual imperfections, due to limited liquidity and moral

hazard, in the relationships between matched entrepreneurs and lenders. Available liquidity

in a given period is limited to the flow of funds to the lender plus currently-produced output.

Payments to entrepreneurs cannot exceed this sum. Further, entrepreneurs must exert effort

to produce output and to maintain their relationships with lenders. In this contracting

environment, the liquidity constraint binds when a lender receives a small flow of funds. The

entrepreneur that is currently matched with the lender cannot then be given the incentive

to exert high effort, and the relationship breaks up. Thus, the combination of liquidity

constraints and moral hazard makes the relationship fragile in the face of fluctuations in

available funds. It follows that the lender’s short run access to liquidity determines whether

the relationship can be sustained in the face of contracting problems.

The number of relationships (lender-entrepreneur matches) is an important state variable

in this economy, since it affects the efficiency of financial intermediation and the returns that

households can earn on investment. The interaction between the number of relationships

and household investment gives rise to multiple Pareto-ranked steady-state equilibria, where

2For example, a series of past loan failures would affect a bank’s equity position, and this in turn would

affect the amount of loans a bank could make, as a consequence of capital requirements. This point is

discussed further in Section 2.4.


positive-activity steady states coexist with a zero-activity “collapse” steady state. Multiple

equilibria emerge from complementarity between intermediation and investment. As the

number of relationships rises, intermediation operates more effectively, and the rate of return

on investment increases. The higher rate of return induces households to provide more

funds in the aggregate, thereby allowing more relationships to be sustained. Aggregate

increasing returns therefore emerge from the process of allocating invested funds via long-

term relationships. For a range of values of aggregate investment, this effect dominates the

usual decreasing-returns effect arising from the production function.

Feedbacks between financial intermediation and investment serve as a mechanism for

propagating aggregate shocks. When an exogenous shock severs a proportion of the re-

lationships, damage is persistent because reforming relationships takes time. In addition,

destruction of relationships causes aggregate investment to fall, due to less efficient inter-

mediation and a consequent decline in investment returns. This, in turn, leads even more

relationships to break up, inducing further reductions in aggregate investment. In this way,

the intermediation-investment complementarity propagates the shock.

If a shock destroys sufficiently many relationships, then the formation of new relationships

can be too slow to offset the ongoing destruction of existing relationships caused by low

investment. In this economic condition, the collapse state becomes the unique equilibrium.

Here collapse is not induced by a sunspot; rather, it becomes unavoidable when the number

of lender-entrepreneur relationships is too low to support adequate investment incentives.

Our theory sheds light on processes that underlie phenomena such as financial collapses

and credit crunches. Outflows of liquidity can damage financial structure by breaking up

credit market relationships, thereby generating further outflows. Timely injections of re-

sources by a policy authority can stave off financial damage, and may prevent the economy

from collapsing.

The results developed here relate to the large literature on “coordination failure,” pi-

oneered by Bryant (1983) and Cooper and John (1988), that stresses the possibility of

low-activity steady states sustained by macroeconomic complementaries. We contribute to


this literature by proposing a novel source of complementaries stemming from frictions asso-

ciated with credit relationships. Since the number of relationships can adjust only gradually,

our approach to coordination failure offers added insights with respect to macroeconomic

dynamics. Importantly, in our model, the low-activity steady state may arise as the unique

equilibrium following a large shock, rather than as one of several equilibria that are condi-

tioned on a sunspot.3

The liquidity-flows approach to credit market frictions, which focusses on external fi-

nance, represents an alternative to established “internal equity” models of credit frictions

that highlight contracting problems created by limited entrepreneurial wealth.4 Both the

liquidity-flows and internal-equity approaches provide mechanisms for propagating macro-

economic shocks, and they can be viewed as complementary perspectives. The liquidity flows

model offers additional predictions pertaining to aggregate increasing returns and multiple

steady states.5

Several other papers have considered how lender wealth constraints contribute to credit

market frictions. Using static models with adverse selection, Farmer (1988a,1988b) shows

that limited access to liquidity can affect the efficiency of contracting and the extent of factor

utilization. Diamond (1984) and Hölmstrom and Tirole (1997a) present microeconomic

3Cooper and Corbae (1997) propose a model of financial collapse based on coordination failure in financial

intermediation. In their paper, households must simultaneously commit to payments in order to finance the

fixed costs of intermediation, and collapse occurs when households believe that other households will not

contribute. Periodic collapse outcomes are tied to a sunspot process.

4Internal equity models based on the costly state verification framework of Townsend (1979) have been

considered by Bernanke and Gertler (1989), Bernanke, Gertler, and Gilchrist (1999), and Carlstrom and

Fuerst (1997). To obtain quantitatively important results, these models rely on specifying very high levels of

audit costs. Kiyotake and Moore (1997) develop a model in which borrower collateral constitutes a binding

constraint on loans, because borrowers are able to “take the money and run.” As pointed out by Berger and

Udell (1995), however, only about half of all lines of credit are secured.

5For simplicity, our model assumes that entrepreneurs do not make use of private wealth for production

or contracting. Our analysis can, thus, also be applied to cases in which the entrepreneur’s wealth is small

relative to the size of the firm.


models that link financial intermediation to the lender’s wealth position. There have been

a number of previous theoretical models of long-term relationships in credit markets; see

Freixas and Rochet (1997, chapter 4). These models have focussed on properties of the

contract between borrower and lender.6

Our analysis draws on formal methods used in the labor literature, particularly Mortensen

and Pissarides (1994) and Ramey and Watson (1997). The modelling of liquidity allocation

to lenders is closely connected to the case of “costly capital adjustment” considered in den

Haan, Ramey and Watson (2000), where the level of a capital input must be chosen before

a relevant shock is realized. For both liquidity allocation and costly capital adjustment, the

key idea is that inputs cannot adjust costlessly.

Finally, Hölmstrom and Tirole (1997b) have considered a model in which low entrepre-

neurial collateral can lead to termination of projects. In their model, financial intermediaries

transfer wealth between entrepreneurs in order to avoid terminations, and aggregate liquidity

can be insufficient when entrepreneurs’ wealth is highly correlated. Our model, in contrast,

considers external finance rather than collateral, and shows that insufficiency of aggregate

liquidity can be brought on by damage to financial structure.

Section 2 presents the model and describes the three key frictions incorporated into the

model. Section 3 lays out the equilibrium conditions. Multiple steady-state equilibria are

derived in Section 4, and propagation of shocks is considered in Section 5. In Section 6,

we highlight the role of allocation frictions; we show that such frictions are critical to the

existence of a collapse steady state. Section 7 concludes.

2. Model

We consider an economy in which there is a single good that may be used for consumption,

investment, and contracting. The agents in this economy are: (i) a representative household;

(ii) a unit mass of intermediaries (whom we call lenders); and (iii) a potentially infinite mass

of entrepreneurs. The agents interact over an infinite number of discrete periods. In each

6Dell’Ariccia and Garibaldi (1998) have recently developed a matching model of bank lending, considering

how matching frictions and breakup costs affect dynamic responses to short-term interest rate shocks.


period, the household makes a consumption/savings decision, whereby “savings” constitute

the quantity of the good supplied to the financial intermediaries. The return on savings

depends on the number of lender-entrepreneur matches and on the process whereby savings

are allocated to lenders.

The lenders form bilateral, long-term relationships with entrepreneurs on a matching

market. Entrepreneurs must pay a fixed posting cost to enter the matching market in a given

period. A lender can only allocate funds received from the household to an entrepreneur

with whom it is currently matched. Thus, an unmatched lender has no outlet for its funds.

In an entrepreneur-lender match, the entrepreneur uses the lender’s funds as a productive

input. Production also requires the entrepreneur to exert effort, which is contractible. Con-

tracting in a given period is resolved according to fixed bargaining weights (as with the Nash

Bargaining Solution). The lender’s gross proceeds are returned to the household at the end

of the period. Further, the lender-entrepreneur relationship is severed if the entrepreneur

exerts low effort in the period; in this case, or if the relationship is severed by agreement,

the agents return to the matching market to find new partners.

Thus, we consider the following three frictions:

• Matching Friction. In a given period, the probability that an unmatched lender orentrepreneur finds a partner is less than one. Moreover, entrepreneurs incur a positive

cost of searching for lenders.

• Allocation Friction. Resources saved by the household are not optimally allocatedamong lenders.

• Contractual Imperfections. Entrepreneur-lender relationships function under limitedliquidity and moral hazard.

The following three subsections describe the model, and the three frictions, in more detail.

Subsection 2.4 provides empirical motivation for the matching and allocation frictions.


2.1. Allocation of Funds. Let Ht denote the aggregate quantity of the good that the

household saves at the end of period t − 1, which is made available for investment at thestart of period t. For simplicity, we assume that the household maximizes the expected

present value of consumption, where r is the discount rate. Let β ≡ 1/(1 + r) denote theimplied discount factor. The household is endowed with a quantity of the good in each

period. We assume throughout that the endowment is sufficiently large to make aggregate

resource constraints nonbinding.

The household is not able to invest in a spot asset market. Rather, investment must flow

through the lenders, who operate on behalf of the household. Let ht denote the quantity

of invested funds received by a particular lender at the start of period t. The allocation of

funds to lenders is subject to random shocks. For this analysis, we abstract from details of

the asset-allocation process, and assume simply that ht is determined by a reduced-form liq-

uidity allocation rule. In our benchmark specification, all lenders (matched and unmatched)

have equal access ex ante to funds; each lender’s allocation ht is drawn according to the

distribution function ν(ht|Ht), assumed to be continuous and increasing in Ht according tofirst-order stochastic dominance. Assume also that ht = 0 is contained in the support of

ν(ht|Ht). Further, total liquidity allocations are equal to the aggregate level:7Z ∞0htdν(ht|Ht) = Ht. (1)

The benchmark allocation rule features two forms of friction, each of which reduces the

efficiency of liquidity allocation and distorts household investment returns: (i) unmatched

lenders get funds, but these lenders cannot offer positive net returns; and (ii) among matched

lenders, allocations may not yield the optimal distribution of funds ex post. In Section 6,

we show that our results are robust to eliminating friction (i), i.e., funds are allocated only

to matched lenders. The results are sensitive, however, to the elimination of both frictions.

7More generally, the liquidity allocation rule could specify that total allocations are less than aggregate

liquidity, reflecting resource costs of allocation. Our results continue to hold in the latter case.


2.2. Lender-Entrepreneur Matching. Lenders invest their liquidity allocations in

projects operated by entrepreneurs. In each period, a lender is either matched in a continu-

ing relationship with an entrepreneur, or else the lender is searching for a new entrepreneur

in whom to invest. Each unmatched entrepreneur may promote a new project by entering

the matching market. To be in the matching pool, an entrepreneur must incur an effort cost

of c > 0 per period. Let Vt denote the mass of entrepreneurs in the matching market in

period t.

The probability that an unmatched lender identifies a promoting entrepreneur in a given

period depends on the scarcity of new projects relative to the total number of unmatched

lenders. Let Ut denote the mass of unmatched lenders at the start of period t, and let

θt ≡ Vt/Ut. A given unmatched lender identifies a new project in period t with probabilityλ(θt), which is continuous, strictly increasing, and satisfies λ(0) = 0. When a new project

is identified, the lender and entrepreneur begin a relationship in the following period. The

aggregate flow of new projects in a period is given by Utλ(θt). The probability that a pro-

moting entrepreneur is matched with a lender, λ(θt)/θt, is assumed to be strictly decreasing

in θt, and satisfies limθt→0 λ(θt)/θt = 1 and limθt→∞ λ(θt)/θt = 0.

2.3. Lender-Entrepreneur Contracting. Lenders and entrepreneurs in continuing re-

lationships negotiate contracts and engage in production in each period. Production requires

both invested funds and effort by entrepreneurs. Entrepreneurial effort is also necessary for

maintenance of the project. We suppose that effort may be either high or low. If the liquidity

allocation is ht and the entrepreneur chooses high effort, then output produced in the period

is given by f(ht). Assume that f(ht) is strictly increasing, strictly concave, and satisfies

f(0) = limht→∞ f0(ht) = 0 and limht→0 f

0(ht) =∞. The choice of high effort further impliesthat the project is maintained, and the relationship continues into the following period.

If low effort is chosen, then zero output is produced, and instead the entrepreneur obtains

a private effort benefit of x > 0. Moreover, low effort causes the project to fail, and the

relationship is severed. In this case, it is assumed that the lender cannot be rematched with

a new entrepreneur until the following period.


The assumption that low effort induces severance of the relationship can be motivated

in a number of ways. To the extent that cooperation is sustained by reputation, a low effort

choice may destroy prospects for future cooperation if renegotiation between the entrepreneur

and lender is costly. Further, even where the agents may renegotiate costlessly, low effort

may give rise to dispute resolution costs that make it too costly to try to preserve the

relationship; see Ramey and Watson (2002). Low effort could also induce a “breakdown”

of productivity that causes the project to produce zero output for some number of periods,

giving the agents an incentive to sever the relationship.8

At the start of the period, the lender and entrepreneur observe the current-period realiza-

tion of ht, and following this they negotiate a contract that determines the division of joint

surplus, along with the entrepreneur’s effort choice. Total contractible wealth for the period

consists of the liquidity allocation plus any output produced. In particular, for simplicity we

assume that the entrepreneur does not have private assets that can be transferred as part of

the contract. The lender is assumed to appropriate the liquidity allocation and output, and

the contract specifies payments to the entrepreneur conditional on his effort choice. Contract

negotiation consists of a first and final offer by the lender, which the entrepreneur may either

accept or reject. If the entrepreneur rejects the offer, then the relationship is severed, and

the lender becomes unmatched. The lender may also opt to sever the relationship in lieu of

making an offer. In either of these cases, the lender may be rematched in the current period.

In equilibrium, the contract will determine a threshold value of ht such that the entre-

preneur chooses low effort, and the relationship breaks up, if and only if ht lies below the

threshold. Letting ht ≥ 0 denote this threshold, the expected value of output, conditionalon Ht, is given by:

µ(ht|Ht) ≡Z ∞ht

f(ht)dν(ht|Ht).

Assume that µ(0|Ht)/Ht is strictly decreasing in Ht. Thus, aggregate returns to scale are8As an alternative to severence of relationships, it can be assumed that low effort leads to persistent pro-

ductivity breakdown. Under this assumption, λ can be specified as the (constant) probability of productivity

recovery. It is straightforward to rework the model and extend the results to this case.


diminishing if ht = 0, i.e., if high effort is always chosen. Our assumptions on f imply that

limHt→0 µ(0|Ht)/Ht =∞ and limHt→∞ µ(0|Ht)/Ht = 0.

2.4. Empirical Motivation for Credit Market Frictions. In this section, we provide

motivation for the matching and allocation frictions specified in our model.9

Motivation for matching friction. The assumption of long-term relationships be-

tween borrowers and lenders is consistent with a large body of literature on financial inter-

mediation. Jaffee and Stiglitz (1990), for example, argue that the market for credit should

be regarded not as an auction market, but rather as a customer market in which borrowers

form relationships with single lenders. As Berlin and Mester (1998) point out, relationship

lending is characterized by close monitoring, renegotiability, and implicit long-term contrac-

tual agreements. In a similar vein, Brewster, Stearns, and Mizruchi (1993) argue that once

a social relation between borrower and lender is established, a company’s freedom to shop

among financial institutions for cheaper sources of capital is constrained. Identifying new

lenders is made more difficult by the tendency of lenders to specialize in particular sectors

or regions.

Direct empirical support for the relationship hypothesis is given by Petersen and Rajan

(1994), who document that firms tend to concentrate their borrowing from one source,

though this concentration decreases as firm size increases. In particular, they find that

the smallest (largest) 10% of the firms in their sample who have a bank as their largest

single lender secure, on average, 95% (76%) of their loans from this bank. In addition

Petersen and Rajan (1994,1995) and Berger and Udell (1995) document that benefits to

borrowers increase with the age of their relationship with banks. For example, Berger and

Udell (1995) show that an additional ten years of bank-borrower relationship lowers the loan

premium by 48 basis points, and lowers the probability of collateral being pledged by about

16 percentage points. Finally, Petersen and Rajan (1994) find that loan rates are smaller for

9The assumptions of limited liability and moral hazard are standard in the literature, and we refer the

reader to Jaffee and Stiglitz (1990) for a discussion of these frictions in credit market transactions.


firms that borrow from only one bank. Indirect evidence for the importance of credit market

relationships is provided by Hoshi, Kashyap, Scharfstein (1990, 1993), Slovin, Sushka, and

Polonchek (1993), Gibson (1995), and Peek and Rosengren (2000), who demonstrate links

between borrowers and the financial health of their banks. Prowse (1998) discusses the

importance of relationship lending for the private equity markets.10

Motivation for allocation friction. Our framework assumes that random factors

influence the allocation of funds across lenders, and that it is costly for lenders to obtain

additional sources of funds at short notice. As a consequence, the realized distribution

of funds might be suboptimal given the needs of different lenders. This subsection offers

empirical examples documenting the importance of such frictions.

The market for commercial and industrial (C&I) bank loans exemplifies the importance

of allocation frictions. In this market, the quantity of bank loans issued is constrained by the

amount of bank equity.11 Importantly, bank equity depends in a large part on past successes

and failures, and obtaining additional bank equity is costly, especially in times of distress.

Since banks often specialize in particular sectors and regions, their loan portfolios are often

not well diversified, which means that banks’ equity positions can display large variations.

The importance of these effects became clear during the banking crises in the mid 1980’s in

Kansas, Oklahoma, and Texas, caused by falling agriculture and energy prices, and during

the banking crises in the early 1990’s in New England tied to falling real estate prices.12

10Trade credit provides another example. If the the seller (the lender in this case) does not have sufficient

funds to extend trade credit to the buyer (the borrower), then the transaction is less likely to take place.

Because the terms of trade credit are typically much less advantageous than for other forms of credit, this

implies that buyers face constraints in other forms of borrowing.

11Implementation of the Basel accord in the Federal Deposit Insurance Corporation Improvement Act

imposes this constraint for the post 1991 period, but even before 1991 legal limits restricted loan sizes

relative to the amount of equity. Moreover, prudent banks limit the size of their loan portfolios relative to

the size of their equity.

12Rose (1993) discusses bank specialization and the implications for diversification. See Amos (1992) and

Bernanke and Lown (1991) for evidence on the mid 1980’s and early 1990’s crises, respectively.


The effect of bank equity on loans has received a great deal of attention in the literature.13

Recent empirical work by Houston, James, and Marcus (1997) and Peek and Rosengren

(1997), which addresses the problem of endogeneity of loan demand, documents that bank

loans are sensitive to internally-generated additions to capital. Negative shocks to capital

therefore limit a bank’s ability to provide its customers with funds.14

The private equity market, discussed by Prowse (1998), gives a second example in which

allocation frictions are important. Intermediaries in the private equity market, called limited

partnerships, identify wealthy investors (for example, institutional investors) who are willing

to provide equity finance to one of the intermediary’s corporate clients. Typically, the

investor requires the limited partnership to also invest in the firm. But since a recent string

of bad outcomes will affect the amount of funds the intermediary can advance, it will also

affect, to some extent randomly, the amount of funds that the intermediary can channel to

its client.

3. Equilibrium

3.1. Equilibrium Contract and Breakup Margin. The equilibrium contract in a

continuing relationship is derived as follows. Let pt denote the payment to the entrepreneur

if high effort is chosen. Clearly, the equilibrium contract will specify that the entrepre-

neur receives nothing if low effort is chosen, since negative payments are not possible. The

13This literature is surveyed by Sharpe (1995).

14A bank could in principle obtain funds by selling C&I loans to other banks, so that the loans would not

be counted against its equity. This mechanism does not appear to counteract the constraining effect of bank

equity in practice, however. Although the market for C&I loans has expanded, it is not as well developed as

is the secondary market for real estate loans, since C&I loans are less standardized. Moreover, the market

for C&I loan sales is dominated by the very large banks: Demsetz (1993) documents that only two sellers

account for an average of 42% of aggregate sales between the first quarter of 1986 and the first quarter of

1992. We mention above that the financial health of a bank affects that of its customers. All of this suggests

that, at least for banks in distress, selling C&I loans does not provide a ready source of relief.


entrepreneur selects high effort if and only if

pt + get ≥ x, (2)

where get indicates the present value of the entrepreneur’s expected future payments if the

relationship continues. Note that the entrepreneur obtains a future return of zero if the

relationship breaks up, since potential entrepreneurs dissipate all the rents from promoting

new projects. Thus, the right-hand side of (2) reflects only current-period private benefit of

low effort.

In negotiating a contract that induces high effort, the lender will offer the smallest value

of pt that satisfies (2). Moreover, pt must be nonnegative, since the entrepreneur has no

assets. Thus, pt is given by

pt = max{x− get , 0}. (3)

The contract is further constrained by the available contractible wealth; i.e., the lender

cannot draw on the future value of the relationship to make current payments to the entre-

preneur. This means that the following liquidity constraint must be satisfied:

f(ht) + ht ≥ pt. (4)

We next consider whether the lender and entrepreneur benefit from continuing the re-

lationship. The lender prefers to offer a contract that induces high effort, as opposed to

severing the relationship at the start of the period, if and only if the following condition

holds:

f(ht) + gt − (pt + get ) ≥ wt, (5)

where gt indicates the present value of expected future joint returns from continuing the

relationship, and wt denotes the present value of the lender’s expected future returns from

entering the pool of unmatched lenders in period t; both terms are net of future liquidity

allocations. The left hand side of (5) constitutes the share received by the lender, consisting

of the the joint returns, f(ht) + gt less current and future payments to the entrepreneur.

Further, the lender will not prefer a contract that induces low effort. Such a contract

would give the lender a current-period net return of zero, while the present value of expected


future returns would be βwt+1, based on commencing search for a new entrepreneur at the

start of the next period. In any equilibrium with positive matching probabilities, this is less

than the return of wt that the lender obtains from breaking up the relationship at the start

of the period. Thus, the lender prefers to continue the relationship if and only if (5) holds.

Condition (2) implies that the entrepreneur always prefers to continue the relationship as

long as the contract induces high effort.

In summary, the relationship continues into the next period if and only if (4) and (5)

hold, where pt is given by (3). If either (4) or (5) are violated, then the relationship breaks

up at the start of the period.

Observe that both constraints (4) and (5) become weaker as ht is increased: f(ht) is an

increasing function of ht; and, conditional on the path of aggregate investment, the future

values gt, get and wt are independent of ht. Thus, there exists a breakup margin ht having the

property that the relationship continues if ht ≥ ht, and breaks up if ht < ht. Combining (3),(4) and (5), it follows that the the following expression determines the breakup margin:15

f(ht) + min{ht −max{x− get , 0}, gt −max{x, get}− wt} = 0. (6)

3.2. Future Returns. The present value of expected future joint returns, gt, is deter-

mined by

gt = β[µ(ht+1|Ht+1) + (1− ν(ht+1|Ht+1))gt+1 + ν(ht+1|Ht+1)wt+1]. (7)

Observe that (7) defines joint returns as a weighted combination of the benefits of continuing

the relationship and the benefits received when the relationship breaks up.

The present value of the entrepreneur’s expected future returns, get , satisfies

get = β

∞Zht+1

(pt+1 + get+1)dν(ht+1|Ht+1). (8)

15In this model, relationships break up because available funds are low. Alternatively, liquidity problems

could be triggered by unexpectedly high random fixed costs, as in Hölmstrom and Tirole (1997b). The key

idea is that lenders face costs to obtain additional funds on short notice.


As for the present value of the lender’s expected future returns from entering the pool of

unmatched lenders, wt, we have

wt = λ(θt)(gt − get ) + (1− λ(θt))βwt+1. (9)

The following lemma derives a constraint on the entrepreneur’s future value that greatly

simplifies the analysis of equilibria.16

Lemma 1. In any equilibrium, get ≤ βx for every t.

The lemma is a consequence of the fact that the entrepreneur’s current payment would

be zero if his future returns exceeded x in any period.

3.3. Matching. Let Nt denote the mass of lenders who enter period t in continuing

relationships. The mass of lenders who seek new projects in period t , Ut, may be expressed

as

Ut = (1−Nt) + ν(ht|Ht)Nt. (10)

The first term on the right-hand side of (10) indicates lenders who enter the current period

unmatched, while the second term captures lenders whose relationships have broken up at

the start of the current period.

The ratio of new projects to unmatched lenders, θt, is determined by the following rent

dissipation condition:λ(θt)

θtget = c. (11)

It is possible to satisfy (11) if c/get ≤ 1; otherwise, θt = 0 holds.The law of motion for the mass of lenders in continuing relationships is given by

Nt+1 = (1− ν(ht|Ht))Nt + Utλ(θt), (12)

where the first term on the right-hand side captures surviving relationships from the pre-

ceding period, and the second term reflects newly-formed relationships.

16The proof of this lemma and the other proofs are given in the appendix.


3.4. Aggregate Investment. Let Rt denote the one-period aggregate net rate of return

on investment:

Rt =

Nt∞Rht

(f(ht)− pt)dν(ht|Ht)Ht

, (13)

where pt is determined by (3) and Ht > 0. For Ht = 0, the aggregate net return is defined

as the limit of (13) as Ht → 0. The representative household demands a rate of return of r.

Thus, Ht gives equilibrium aggregate investment if

Rt = r. (14)

Additionally, if Rt < r at Ht = 0, then Ht = 0 gives an equilibrium value.

To summarize, given an initial number of relationships N1, equations (6)-(14) jointly

determine ht, gt, get , wt, Ut , θt, Nt+1, Rt and Ht for t = 1, 2, ... .

4. Multiple Steady States

In this section we demonstrate that the model possesses multiple steady-state equilibria

(SSE), associated with differing levels of aggregate activity. Making use of Lemma 1, steady-

state versions of equilibrium conditions (6)-(12) may be written as follows:

f(h) + min{h− x+ ge, g − x− w} = 0, (15)

g = β[µ(h|H) + (1− ν(h|H))g + ν(h|H)w], (16)

ge = β(1− ν(h|H))x, (17)

w = λ(θ)(g − ge) + (1− λ(θ))βw, (18)

U = (1−N) + ν(h|H)N , (19)

1

θλ(θ)ge = c if

c

ge≤ 1, θ = 0 if

c

ge> 1, (20)

N = (1− ν(h|H))N + Uλ(θ). (21)

Finally, the steady-state aggregate net return may be written:

R =N

H[µ(h|H)− (x− ge)(1− ν(h|H)]. (22)


Equilibrium aggregate investment satisfies the following:

Either R = r, or H = 0 and R < r. (23)

We now state conditions under which there exists a SSE with H > 0, meaning that there

is productive activity in the economy. The household is willing to choose positive aggregate

investment only if it earns a sufficiently high return. From (22), it may be seen that the

return depends on the size of payments to the entrepreneur: the return may be too low if

x−ge is too high. Since payments to the entrepreneur are based on x, it follows that small xis needed to sustain a SSE with H > 0. According to (17), however, potential entrepreneurs

have little incentive to promote new projects when x is low. As a consequence, for given

values of x, we must specify values of c that are sufficiently low to support θ > 0, and thus

N > 0, in equilibrium.

It follows that low values of x must be accompanied by sufficiently low values of c in

order to ensure that θ > 0. The following proposition uses a tractable condition on c that is

sufficient for this.

Proposition 1. Choose any ω ∈ (0, 1), and let c be given by c = ωβx. If x is sufficiently

small, then there exists a SSE with H > 0.

The proof makes use of the fact that when x = 0, there exists a unique SSE, having strictly

positive aggregate investment. The proof constructs an equilibrium that is a perturbation

of this SSE. The assumption c = ωβx is used to rule out the possibility that N collapses to

zero when x is small.

In the positive-activity equilibrium of Proposition 1, h > 0 will hold as a consequence of

x > 0, and thus there is a positive probability that a low liquidity allocation will cause a

given continuing relationship to break up in any period. Contracting problems, in the form of

liquidity constraints combined with entrepreneurial moral hazard, cause credit market rela-

tionships to be fragile in the face of fluctuations in available liquidity. This financial fragility

is manifested in the positive failure rate of established relationships, offset in equilibrium by

the initiation of new projects.


When x is strictly positive, there will also exist a SSE in which H = 0, meaning that

all economic activity ceases. The following proposition establishes the existence of this

“collapse” equilibrium.

Proposition 2. If x > 0 then there exists a SSE with H = 0.

In the collapse equilibrium of Proposition 2, entrepreneurs have no incentive to promote

new projects, since no funds are available. As relationships are not created, the household

obtains a zero return on any funds given to the lenders, and so nothing is invested. The key

credit market imperfection in this case is that the household cannot channel funds directly to

new entrepreneurs; rather, funds must flow through lenders, subject to allocation frictions.

Propositions 1 and 2 together imply that the model possesses multiple SSEs for small

positive values of x: a positive-activity equilibrium coexists with the collapse equilibrium.

The existence of multiple equilibria reflects complementarity between the structure of finan-

cial intermediation, represented by the number of continuing relationships, and investment.

High values of N bring about more efficient intermediation, leading to high values of R and

a positive choice of H by the household. Low values of N , in contrast, cause intermediation

to operate less effectively. Since R is low as a consequence, the household does not wish to

invest, and the collapse outcome obtains.

The relationship between aggregate investment and financial intermediation is illustrated

in Figure 1. For convenience, the example uses a liquidity allocation rule having a two-point

support, with one of the points being ht = 0.17 The figure reports the values of the number of

17For the examples, the support of ν(ht|H) is taken to consist of 0 and hu(H), where ν(hu|H) ≡min{[H/(H + γ1)]γ0(H + γ2)

η, 1}, η < 1, and hu(H) ≡ H/ν(hu|H). Although our specification does

not satisfy the assumption that ν(ht|H) should be continuous, the discrepency is inessential, since wecan regard the specification as approximating a continuous distribution function that increases sharply

at h = 0 and h = hu(H). The production function is given by f(ht) = hαt and the matching function by

m(Ut, Vt) = ξUζt V

1−ζ . The examples use the parameters x = 3, α = 0.3, β = 0.96, ξ = 0.25, ζ = 0.5,

c = 0.342, η = 0.2, γ0 = 0.401, γ1 = 0.0001, and γ2 = 0.0003. For the values of γ1 and γ2 considered, the

results are numerically indistinguishable from those in which ν(hu|H) = γ0Hη, which is the allocation rule


Effect of investment on steady-state values of financial intermediation

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 10 20 30 40 50 60 70 80 90

aggregate investment

breakup probabilities

number of relationships

matching probability

Figure 1:

relationships, N , the lender matching probability, λ(θ), and the breakup probability, ν(h|H),that emerge from solutions to (15)-(21) for given values of aggregate investment, H. Note

first that, for a region of very low H, no relationships form; thus, financial intermediation

cannot occur at all unless aggregate investment exceeds a minimum threshold. For a range

of high values of H, the equations have a single solution, with positive N and λ(θ), and

values of ν(h|H) lying below unity. In this case, an increase in H leads to a larger numberof relationships, a higher matching probability, and a lower breakup probability. Finally, for

a middle range of H, positive-valued solutions coexist with zero-valued solutions.18

used in earlier drafts of this paper. In the extension of Section 6, however, we need this slight modification

to satisfy the sufficiency conditions. In order to compare the results for different environments, we use the

modified allocation rule in this section as well.

18For these middle values of H, a value of ge equal to zero implies that relationships break up even at

h = hu(H), and so ge indeed equals zero. At the same time, assuming that no breakup takes place at

h = hu(H) implies a value for ge such that in fact no break up occurs at h = hu(H). This accounts for

the coexistence of zero and positive solutions. Below we will select the positive steady states in cases where


Effect of liquidity on average returns

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 10 20 30 40 50 60 70 80 90


"x = 0" economy

"fragile" economy(all lenders can get funds)

"fragile" economy(only matched lenders can get funds)

Figure 2:

The solid line in Figure 2 plots the values of the average return, R, associated with

the solutions in Figure 1; average returns are derived using (22). The figure illustrates the

existence of multiple SSE. The region of low H for which R = 0 may be noted; the ori-

gin indicates the collapse equilibrium derived in Proposition 2. For a middle region of H,

average returns rise with H, as the favorable effect of higher liquidity on the number of

continuing relationships outweighs the effect of diminishing returns within individual rela-

tionships. Complementarity between H and N thus generates aggregate increasing returns

on this region. For high H, diminishing returns come to dominate. These conflicting effects

give rise to a pair of equilibria with positive H. Returns for the x = 0 economy, in which

all lenders are matched in every period, are given by the upper curve. In contrast to the

fragile economy, the x = 0 economy exhibits diminishing returns for all H and has a unique

steady-state equilibrium. The line labeled “only matched lenders can get funds” is discussed

in Section 6.

multiple solutions exist.


5. Propagation of Shocks

This section shows how complementarity between the structure of financial intermediation

and aggregate investment helps to propagate aggregate shocks. To illustrate how shocks are

propagated, we consider a shock that takes the form of a surprise reduction in the number

of continuing relationships. This may be interpreted as a negative productivity shock. The

economy is assumed to enter period 1 in a positive-activity SSE, with N giving the number

of continuing relationships. Before the level of aggregate liquidity is determined, however,

N drops to a lower level, reflecting exogenous breakup of relationships.19

Figure 3 presents numerically calculated equilibrium values associated with a negative

shock to the number of relationships, aggregate investment, and the breakup probabil-

ity.20Observe that the number of relationships drops sharply, but returns only gradually

to its SSE value; thus, the shock has a persistent effect on the structure of intermediation.

The need for lenders to gradually rematch is one source of this persistence. As documented

by the graph, there is also a large and persistent decline in aggregate investment follow-

ing the shock, reflecting the investment response to lower levels of Nt and higher levels of

ν(ht|Ht). Correspondingly, the breakup probability ν(ht|Ht) remains persistently above itssteady-state level, further slowing the return of Nt to the steady state.

The resulting effects on output are shown in Figure 4, which compares output in the

equilibrium (thick solid line) to the path that would emerge if Ht were held fixed at its

SSE level (thin solid line), so that propagation would be driven solely by lender rematching.

Observe that intermediation-investment feedbacks serve to magnify the shock on impact,

and overall they roughly double the output loss in this example.

The credit market response to the shock involves two competing effects. On one hand,

19Alternatively, one can consider an experiment where aggregate productivity is unexpectedly lower in

period zero but has recovered in period one. The results of this experiment are similar, except that because

of the burst of destruction in period zero, N1 will lie below the steady state level N at the outset of period

one.

20Details are given in footnote 17.


Effect of financial structure shock

0.045

0.046

0.047

0.048

0.049

0.05

0.051

0.052

0.053

0.054

0.055

-1 0 1 2 3 4 5 6 7 8 9 10

period

0.975

0.98

0.985

0.99

0.995

1

1.005

1.01

1.015

1.02

1.025

breakup probability


number of relationships

Figure 3:

Effect of financial structure shock on output

0.975

0.98

0.985

0.99

0.995

1

-1 0 1 2 3 4 5 6 7 8 9 10

period

only matched lenders can get fundsall lenders can get funds

"fixed aggregate investment" economies

"endogenous aggregate investment" economies

Figure 4:


structure is repaired via matching, as relationships are reformed. On the other hand, adverse

feedbacks raise the rate at which relationships break up along the adjustment path. The

latter effect can dominate, so that the market becomes unable to escape the collapse outcome.

The next proposition gives conditions under which this situation can arise.

Proposition 3. If c lies sufficiently close to βx, then for N1 sufficiently small, there is a

unique equilibrium, having Ht = Nt+1 = 0 for all t.

The key point is that collapse of the credit market emerges as the unique equilibrium

for a sufficiently large shock to the structure of financial intermediation. In contrast to the

existing literature, coordination failure in this case does not entail equilibrium selection or

sunspot arguments that serve to align agents’ expectations. What happens instead is that

a big shock does so much damage to financial structure that recovery becomes impossible.

Proposition 3 identifies a condition on the entrepreneur’s project promotion cost, c, that is

sufficient to ensure that the rate at which relationships are rematched is too slow to offset

the ongoing increase in their destruction due to investment-intermediation feedbacks.21 The

collapse outcome becomes an absorbing state, and only a sustained exogenous injection of

liquidity can restore credit market activity.

6. Role of Allocation Frictions

This section analyzes in more detail the role of allocation frictions in the model. The bench-

mark liquidity allocation rule from Section 2 embodies two kinds of frictions: (i) unmatched

lenders obtain funds; and (ii) allocations to the matched lenders may yield a suboptimal

distribution of funds ex post. In this section we consider the implications of eliminating

these frictions. Our results continue to hold if the liquidity allocation rule gives funds only

21It should be noted that steady-state equilibria with positive investment can exist under conditions

supporting Proposition 3. In particular, assume that for given ht, ν(ht|Ht) may be made arbitrarily smallby taking Ht sufficiently large. Then positive-investment equilibria will exist under the conditions of the

theorem as long as f(ht) is sufficiently large for ht outside of a neighborhood of zero, i.e., for a sufficiently

high level of productivity.


to matched lenders, so that friction (i) is absent. If both frictions are eliminated, however,

then the collapse equilibrium need no longer be a robust outcome of the model: an alloca-

tion rule can be found such that, as long as N1 > 0, every equilibrium acheives an efficient

allocation in the limit.22

6.1. Allocation to Matched Lenders Only. In this subsection, we describe a class

of liquidity allocation rules that give funds only to matched lenders, and under which our

propositions continue to hold. Let the liquidity allocation formatched lenders be determined

according to the continuous distribution function ν(ht|Ht, Nt) which is now conditioned onthe number of matched lenders as well as on aggregate liquidity. Unmatched lenders receive

no funds. For fixed Ht, ν(ht|Ht, Nt) is assumed to be decreasing in Nt according to first-order stochastic dominance, reflecting the need to spread the given quantity of aggregate

investment over a greater number of matched firms. The expected value of production, given

the breakup margin ht and conditional on Ht and Nt, is defined by:

µ(ht|Ht, Nt) ≡Z ∞ht

f(ht)dν(ht|Ht, Nt).

For fixed Nt, the functions ν(ht|Ht, Nt) and µ(0|Ht, Nt) are taken to satisfy the earlierassumptions. In particular, total allocations are restricted as follows:

Nt

Z ∞0htdν(ht|Ht, Nt) = Ht

Further, the following is assumed to hold for each y and z satisfying 0 < y < z:

lim(Ht,Nt)→(0,0)

Nt

Z z

y

htHtdν(ht|Ht, Nt) = 0. (24)

Thus, individual allocations in intermediate ranges of total allocations become unlikely when

Ht and Nt are low. The key friction here is that the financial system becomes unbalanced

between small and large lenders when the scale of intermediation is small.

The proofs of Propositions 1 through 3 extend readily to this alternative specification of

the allocation rule. Positive-investment equilibria with low Ht cannot be sustained when Nt

22Inefficiencies due to contracting frictions would remain, given that x > 0. See note 26 below.


is small, because matched lenders obtain either very low allocations, insufficient to sustain

their relationships with entrepreneurs, or very high allocations, making them inefficiently

large. The resulting distribution of funds yields an aggregate return that is insufficient to

induce the household to invest.23

In Figure 2, the dashed line depicts the average return R as a function of H for an

example that satisfies the assumptions of this subsection.24 Aggregate increasing returns

and multiple steady states, including a collapse equilibrium, continue to exist even when all

funds go to matched lenders. For this case, Figure 4 plots the effects of a negative aggregate

shock on equilibrium output (thick dashed line) and on the path that would emerge if Ht

would be held constant (thin dashed line). As demonstrated by the figure, propagation of

the aggregate shock is qualitatively similar to the benchmark. This case behaves similarly to

the benchmark because, in each period, a proportion of matched lenders obtains insufficient

liquidity to maintain relationships with entrepreneurs. These unlucky lenders sever their

relationships and yield zero net returns, as would unmatched lenders who obtain funds in

our benchmark setting.25

6.2. Balanced Allocations. What underlies the collapse outcome is that when aggre-

gate investment and the number of relationships are low, relatively few matched lenders

receive allocations that both sustain incentives and generate adequate returns. If the alloca-

tion rule could better discriminate among subsets of lenders, giving some lenders moderate

proportions of available funds and others zero funds, then it would become possible to pre-

serve a small number of relationships in the face of low aggregate investment. Such balanced

allocation rules make it possible to avoid the collapse outcome and bring about an optimal

23It can be shown that under our assumptions, (24) holds uniformly in Ht as Nt → 0. Thus, small Nt will

suffice to rule out positive-investment equilibria at all levels of Ht, as needed for the proof of Proposition 3.

24The allocation rule is identical as the one given in footnote 17, but now hu ≡ (H/N)/ν(hu|H).25Although the effects in the figures appear quantitatively smaller under the new rule, it is important to

note that we did not recalibrate the parameters. For example, steady-state breakup levels are lower for the

economy in which only matched lenders obtain funds.


distribution of funds in the SSE.

Consider the following nonrandom allocation rule. IfHt ≥ (1−β)xNt, then each matchedlender receives ht = Ht/Nt. IfHt ≤ (1−β)xNt, then the allocation ht = (1−β)x is given to asubset of matched lenders having massHt/(1−β)x, and the remaining lenders obtain ht = 0.Further, in the latter case the allocation rule gives priority to continuing matches over newly-

formed matches. That is, if Ht ≥ (1 − β)xN 0t, where N

0t ≤ Nt is the mass of relationships

that continue from the previous period, then each of these continuing relationships receives

a positive allocation. Observe that this rule violates (24), since the average value of ht/Ht is

unity for all Ht ≤ (1−β)xNt. Note that as Ht approaches zero, the allocations are balanced

among a vanishingly small number of firms.

Under this balanced allocation rule, lender-entrepreneur matches are able to continue

indefinitely as long as pt ≥ (1−β)x for all t, which is indeed the case given the rule and the

assumption that x is sufficiently small. Thus, as long as the effort cost for promoting new

projects is sufficiently low relative to x, the number of matches can only grow over time. To

state this formally, we have:

Proposition 4. If c < βx and x is sufficiently small, then there is a liquidity allocation

rule that violates (24) such that, for any N1 > 0, every equilibrium has Nt → 1 over time.

In the unique SSE of this economy, all lenders are matched with entrepreneurs. In view

of the strict concavity of the production function f , it is efficient to give the same level of ht

to each lender, and so the distribution of funds in the steady-state equilibrium is optimal.26

Real-world implementation of such an allocation rule, however, would clearly place heavy

demands on the informational and contractual capabilities of the financial sector. Frictions

in allocating liquidity, modelled here in terms of randomness and unevenness of allocations

26Aggregate investment is inefficiently low in the SSE, however, since contracting frictions imply that

entrepreneurs must earn positive surplus in order to provide incentives for high effort. The balanced allo-

cation rule achieves the most efficient allocation of aggregate investment across lenders given this incentive

constraint.


across lenders, reflect plausible limitations on these capabilities, and have important impli-

cations for the functioning of the capital market.

7. Conclusion

Long-term relationships between borrowers and lenders are a common feature of credit mar-

ket trading. This paper considers a model wherein relationships constitute channels through

which invested funds flow from households to entrepreneurs. Relationships become liquidity

constrained when lenders receive low flows, causing them to break up. As the number of

relationships falls, financial intermediation becomes less efficient, and the returns earned by

households decline. Thus, financial intermediation and investment are complementary in

the aggregate. Because of this complementarity, multiple steady-state equilibria may ex-

ist, including a low activity “collapse” outcome. For a sufficiently large shock to financial

structure, collapse becomes the unique equilibrium.

Our results have implications for policy responses to financial crises. Importantly, out-

flows of capital associated with crises may cause lasting damage to financial structure, in-

hibiting any subsequent inflows by making investment less attractive. These feedbacks to

financial structure make for slower recovery from crisis episodes, and may drive economies

into persistent low-activity states. Policy authorities can potentially prevent financial dam-

age through interventions designed to support aggregate investment or sustain credit market

relationships. Such interventions are motivated by the need to preserve valuable channels

through which external investment may flow back into the economy. Policy delays can be

costly in this context, since damage continues as long as aggregate investment remains low.

The model relies on a number of simplifying assumptions that make the analysis tractable,

but that are not essential for our main conclusions. The process of liquidity allocation, in

particular, could be analyzed in greater detail. Households may invest effort in directing

asset flows toward more favorable lenders, for example. Lenders may attempt to swap

liquidity after observing their allocations and their success at locating new projects. The

key assumption, however, is that such processes do not eliminate all allocation errors. As

the structure of intermediation weakens, errors rise, leading to lower returns and reduced


investment, thereby weakening structure further. Thus, the fundamental complementarity

driving the results is robust to allowing for richer liquidity allocation structures.

The assumption that entrepreneurs do not use personal wealth for investment or con-

tracting may also be weakened without undermining the main results. In such an extended

model, internal and external finance would combine to determine the breakup margin. High

reliance on internal finance can be viewed as a mechanism for insuring against the fragility

problems that we highlight.

>From the standpoint of modern economies like the U.S., the importance of the frictions

highlighted in this paper depends on the particular financial market considered, and also on

the period in question, since financial institutions have gone through remarkable changes.

The development of the commercial paper market in the 1960’s, and the elimination of

restrictions on interstate banking after the implementation of the Riegle-Neal Interstate

Banking and Branching Efficiency Act in 1997, are likely to enhance the efficiency of liquidity

allocation. On the other hand, the recent implementation of the Basle Accord, restricting

the amount of loans a bank can issue relative to the bank’s equity position, may exacerbate

allocation frictions, given that it is costly for banks to attract additional equity. Moreover,

adjustments in retained earnings are unlikely to counteract this effect and bring about an

efficient allocation of equity across banks.

Another important development has been the recent consolidation in the banking sector,

resulting in fewer banks and the emergence of very large banks. This may improve the

efficiency of allocation, if the internal allocation of funds within big banks is efficient. Some

authors, however, have pointed out that large banks might be less interested in C&I loans

to small firms,27 aggravating the matching friction, while the loss of personal relationships

between entrepreneurs and owners or presidents of regional banks might worsen moral hazard

problems. Finally, in the recently developed markets for private equity, as well as the markets

for venture equity and trade credit, one can expect allocation frictions to remain relevant.

27Drabenstott and Meeker (1997) point out: “When rural banks are acquired by metropolitan banks, some

customers find the acquired bank is no longer interested in serving small local businesses”.


Assessing more completely the empirical significance of allocation frictions is an important

topic for future research.


APPENDIX

Proof of Lemma 1. From (3), we see that the most an entrepreneur can obtain in any

single period is x. Thus, get ≤P∞k=1 β

kx = βx/(1 − β). Suppose there is a contingency

(including realizations of ht and other variables describing the economy) under which get > α,

for some α > βx. From (8) it follows that there must be a contingency in this relationship,

occurring with positive probability, for which pt+1 + get+1 > α/β. Note that (3) implies

pt+1+ get+1 = max{x, get+1}. If pt+1+ get+1 = x, then we have x > α/β, contradicting βx < α;

thus, pt+1 + get+1 = get+1 > α/β. Iterating this argument, observe that for each k there is a

contingency occurring in period t+ k under which get+k > α/βk. This yields a contradiction

for large k. Q.E.D.

Proof of Proposition 1. Equations (17), (19), (20), (21) and c = ωβx may be combined

into the following two equations:

λ(θ)

θ(1− ν(h|H) = ω if

ω

1− ν(h|H) ≤ 1, θ = 0 otherwise. (25)

N [ν(h|H)(1− λ(θ)) + λ(θ)] = λ(θ), (26)

(25) and (26) have a unique solution (θ, N) = (θE(h,H), NE(h,H)) that is continuous in h

and H. Moreover, since ω < 1, it follows that θE(h,H) > 0 for h sufficiently close to zero.

Now define the functions ψA(h,H, x) and ψB(h,H, x) by

ψA(h,H, x) ≡ f(h) + min{h− (1− β(1− ν(h|H))x,

(1− λ(θE))βµ(h|H) + λ(θE)β(1− ν(h|H))x1− β(1− ν(h|H))(1− λ(θE))

− x},

ψB(h,H, x) ≡ NE

H{µ(h|H)− [1− β(1− ν(h|H))](1− ν(h|H)x}.

The function ψA is obtained by substituting (16)-(18) into (15). For ψB, (17), (19) and (21)

are substituted into the first part of (23).


Lemma 2. Suppose h and H satisfy the following conditions:

ψA(h,H, x) = 0, (27)

min{r − ψB(h,H, x),H} = 0. (28)

Then there exists a SSE having equilibrium values h and H.

Proof of Lemma 2. Let g − w be given by

g − w = (1− λ(θE))βµ(h|H) + λ(θE)β(1− ν(h|H))x1− β(1− ν(h|H))(1− λ(θE))

. (29)

Let (θ, N) = (θE(h,H), NE(h,H)), and let ge and U be determined by (17) and (19),

respectively. Then (27) implies (15), while (28) assures that (23) is satisfied. Using (29) it is

possible to define g and w that satisfy (16) and (18). Finally, (17) and (19) together imply

(20) and (21). Q.E.D.

Now let hL(H, x) denote the smallest value of h that satisfies

f(hL) + hL − (1− β(1− ν(hL|H)))x = 0. (30)

Thus, (15) and (17) together ensure that h ≥ hL(H, x) in any SSE. Note that x > 0 impliesthat hL(H, x) > 0 for all H ≥ 0, while limx→0 hL(H,x) = 0. Further, for a given interval

[H l, Hu], taking x to be sufficiently small assures that the left-hand side of (30) is strictly

increasing in hL for all H ∈ [H l,Hu], and thus hL is a continuous function of H on [H l,Hu].

Define the following:

νL(H, x) ≡ ν(hL(H,x)|H).

This is a continuous function of H on [H l,Hu] for x sufficiently small, based on continuity of

hL. In addition, limx→0 hL(H,x) = 0 implies limx→0 νL(H,x) = 0, limx→0NE(hL(H,x), H) =

1 and limx→0 θE(hL(H, x),H) > 0 for all H; let θ0 ≡ limx→0 θE(hL(H,x),H).Next, let H∗ be the unique solution to

µ(0|H∗)H∗ = r. (31)


Note that H∗ > 0. Choose values H l ∈ (0,H∗) and Hu > H∗. Suppressing arguments, we

have, for all H ∈ [H l, Hu]:

limx→0

(1− λ(θE))βµ(hL|H) + λ(θE)β(1− νL)x

1− β(1− νL)(1− λ(θE))=(1− λ(θ0))βµ(0|H)1− β(1− λ(θ0))

> 0.

This implies, for sufficiently small x:

hL − (1− β(1− νL))x <(1− λ(θE))βµ(hL|H) + λ(θE)β(1− νL)x

1− β(1− νL)(1− λ(θE)). (32)

Moreover, continuity of the functions hL, νL and θE(h,H) means that (32) will hold for all

H ∈ [H l,Hu] when x is small. Applying (30) and (32), it follows that for sufficiently small

x, ψA(hL, H, x) = 0 for all H ∈ [H l,Hu]. Thus, (27) is satisfied.

Finally, observe that

limx→0ψ

B(hL, H, x) =µ(0|H)H

.

Under our assumptions, we have

µ(0|H l)

H l> r >

µ(0|Hu)

Hu.

Thus, the following is obtained when x is sufficiently small:

ψB(hL,H l, x) > r > ψB(hL,Hu, x).

Letting x be small enough to assure that ψB(hL, H, x) is continuous in H on [H l, Hu], it

follows that ψB(hL,H 0, x) = r for some H 0 ∈ [H l, Hu]. Thus, (28) holds. Setting h =

hL(H 0, x) and H = H 0, and invoking Lemma 2, we obtain a SSE with H > 0. Q.E.D.

Proof of Proposition 2. Fix x > 0 and suppose that infH≥0 hL(H, x) = 0. Take a sequence

{Hn}∞n=1 with limn→∞ hL(Hn, x) = 0.

limn→∞{f(h

L(Hn, x)) + hL(Hn, x)− (1− β(1− ν(hL(Hn, x)|Hn))x}

= −(1− β)x < 0,

which contradicts the condition (30) that defines hL(H,x). Thus, infH≥0 hL(H,x) > 0. Let

h0 ≡ infH≥0 hL(H, x).


Using (22) and (1), we have, for all h ≥ h0 and all N :

ψB(h,H, x) ≤ NHµ(h|H) = N

H

Z ∞hf(h)dν(h|H)

≤ f(h)hNZ ∞h

h

Hdν(h|H) ≤ f(h

0)h0

NZ ∞h0

h

Hdν(h|H) < f(h0)

h0N.

Let N 0 > 0 satisfyf(h0)h0

N 0 < r. (33)

It follows that r − ψB(h,H, x) < 0 for all h ≥ h0, N < N 0. Letting H = 0, we have

ν(h0|0, N) = 1, whence (θE(h, 0), NE(h, 0)) = (0, 0) for all h ≥ h0. Choosing h to satisfyψA(h, 0, x) = 0 completes the proof, in view of Lemma 2. Q.E.D.

Proof of Proposition 3. Applying Lemma 1 to (3) and (8), we have get = β(1−ν(ht|Ht))x.Thus, the following holds in any equilibrium:

min{ht − x+ get , gt − x− wt} ≤ ht − (1− β)x.

Let h0 be defined by

f(h0) + h0 − (1− β)x = 0.

It follows that h0 > 0, and also ht ≥ h0 must hold in equilibrium.Using (13) and (1), we have, for all Nt and Ht:

Rt ≤ NtHt

Z ∞ht

f(ht)dν(h|Ht) ≤ f(h0)

h0Nt

Z ∞h0

htHtdν(ht|Ht) < f(h0)

h0Nt. (34)

Let N 0 > 0 satisfy (33). If N1 < N 0 , then (34) implies R1 < r for all H1, and thus the only

value consistent with equilibrium is H1 = 0.

Next, condition (11) may be written

λ(θt)

θt=

c

β(1− ν(ht|Ht))x.

As long as c/βx is sufficiently close to unity, we can be sure that θt lies as close to zero as

desired, so that λ(θt) < N 0 holds.


Now observe

h1(1− ν(h1|0)) ≤Z ∞0h1dν(h1|0) = 0,

so that ν(h1|0) = 1. This implies U1 = 1 and N2 = λ(θ1) < N 0. In turn, N2 < N 0

implies H2 = 0, by the argument used above. Thus, ν(h2|H2) = 1, and in fact θ1 = 0; thenN2 = H2 = 0 is implied. By induction, this result can be extended to all t. It may be

verified, as in the proof of Proposition 2, that these values give an equilibrium. Q.E.D.

Proof of Proposition 4. Suppose a lender-entrepreneur relationship never breaks up once

it is formed. In this case, the entrepreneur’s continuation value satisfies:

get = β∞Xs=1

βs−1pt+s (35)

Further, pt = x−get must hold for all t, else pt > x−get ≥ 0 for some t, and the lender wouldoffer a lower value of pt. Together with (35) this implies get = βx and pt = (1− β)x for all

t. In fact, if ht ≥ (1 − β)x is assured for all t, the lender and entrepreneur will choose to

continue their relationship in every period.

With this, the aggregate net rate of return is given by

Rt =

f((1−β)x)(1−β)x − 1, if Ht ≤ Nt(1− β)x,

NtHt[f(Ht

Nt)− (1− β)x], if Ht > Nt(1− β)x.

The following holds for sufficiently small x:

f((1− β)x)

(1− β)x− 1 > r,

and we have that Ht > Nt(1 − β)x for all t. This means aggregate investment is sufficient

to sustain existing relationships, and relationships never break up.

Finally, let θ0 be defined byλ(θ0)θ0

βx = c.

Since get = βx, taking c sufficiently small ensures that θ0 > 0, whence (11) implies θt = θ0

for all t. From (12) the law of motion for Nt becomes

Nt+1 = (1− λ(θ0))Nt + λ(θ0),

and Nt → 1 is implied. Q.E.D.


References

[1] Amos, O.M. Jr. (1992), “The Regional Distribution of Bank Closings in the United

States from 1982 to 1988,” Southern Economic Journal 58, 805-815.

[2] Berger, A.N., and G.F. Udell (1995), “Relationship Lending and Lines of Credit in

Small Firm Finance,” Journal of Business 68, 351-381.

[3] Berlin, M., and L.J. Mester (1998), “On the profitability and cost of relationship lend-

ing,” Journal of Banking and Finance 22(6-8), 873-897.

[4] Bernanke, B., and M. Gertler (1989), “Agency Costs, Net Worth, and Business Fluctu-

ations,” American Economic Review 79, 14-31.

[5] Bernanke, B., M. Gertler, and S. Gilchrist (1999), “The Financial Accelerator in a

Quantitative Business Cycle Framework”, in J. Taylor and M. Woodford (eds.), The

Handbook of Macroeconomics, Amsterdam: North-Holland.

[6] Bernanke, B., and C.S. Lown (1991), “The Credit Crunch,” Brookings Papers on Eco-

nomics Activity 2, 205-248.

[7] Brewster Stearns, L., and M. S. Mizruchi (1993), “Corporate Financing: Social and

Economic Determinants,” in: Explorations in Economic Sociology, R. Swedberg (ed.),

Russell Sage Foundation, New York, 278-307.

[8] Bryant, J. (1983), “A Simple Rational-Expectations Keynes-Type Model,” Quarterly

Journal of Economics 98, 525-528.

[9] Carlstrom, C.T., and T.S. Fuerst (1997), “Agency Costs, Net Worth, and Business

Fluctuations: A Computable General Equilibrium Analysis,” American Economic Re-

view 87, 893-910.

[10] Cooper, R., and A, John (1988), “Coordinating Coordination Failures in Keynesian

Models,” Quarterly Journal of Economics 103, 441-463.


[11] Cooper, R., and D. Corbae (1997), “Financial Fragility and the Great Depression,”

Journal of Economic Theory, forthcoming.

[12] Dell’Ariccia, G., and P. Garibaldi (1998), “Bank Lending and Interest Rate Changes in

a Dynamic Matching Model,” IMF Working Paper WP/98/93, June.

[13] Demsetz, R. (1993), “Recent Trends in Commercial Bank Loan Sales,” Federal Reserve

Bank of New York Quarterly Review 18, 75-78.

[14] Den Haan, W.J., G. Ramey, and J. Watson (2000), “Job Destruction and Propagation

of Shocks,” American Economic Review 90, 482-498.

[15] Diamond, D. (1984), “Financial Intermediation and Delegated Monitoring,” Review of

Economic Studies 51, 393-414.

[16] Drabenstatt, M., and L. Meeker (1997), “Improving Rural Financing,” Independent

Banker July, 33-35.

[17] Farmer, R. (1988a), “What is a Liquidity Crisis?” Journal of Economic Theory 46,

1-15.

[18] Farmer, R. (1988b), “Money and Contracts,” Review of Economic Studies 55, 431-446.

[19] Freixas, X., and J.-C. Rochet (1997), Microeconomics of Banking, Cambridge, MA:

MIT Press.

[20] Gibson, M.S. (1995), “Can Bank Health Affect Investment? Evidence from Japan,”

Journal of Business, 68, 281-308.

[21] Hölmstrom, B., and J. Tirole (1997a), “Financial Intermediation, Loanable Funds, and

the Real Sector,” Quarterly Journal of Economics 112,

[22] Hölmstrom, B., and J. Tirole (1997b), “Private and Public Supply of Liquidity,” Journal

of Political Economy 106, 1-40.


[23] Hoshi, T., A.K. Kashyap, and D.S. Scharfstein (1990), “Bank Monitoring and Invest-

ment: Evidence from the Changing Structure of Japanese Corporate Banking Rela-

tionships,” in R.G. Hubbard, ed., Asymmetric Information, Corporate Finance, and

Investment, Chicago: University of Chicago Press.

[24] Hoshi, T., A.K. Kashyap, and D.S. Scharfstein (1993), “Corporate Structure, Liquidity

and Investment: Evidence from Japanese Industrial Groups,” Quarterly Journal of

Economics 106, 33-60.

[25] Houston, J., C. James, and D. Marcus (1997), “Capital Market Frictions and the Role

of Internal Capital Markets in Banking,” Journal of Financial Economics 46, 135-164.

[26] Jaffee, D., and J. Stiglitz (1990), ”Credit Rationing,” in B.H. Friedman and F.H. Hahn,

eds., Handbook of Monetary Economics, Volume II, Amsterdam: Elsevier, 837-888.

[27] Kiyotaki, N., and J. Moore (1997), “Credit Cycles,” Journal of Political Economy 105,

211-248.

[28] Mortensen, D.T., and C.A. Pissarides (1994), “Job Creation and Job Destruction in the

Theory of Unemployment,” Review of Economic Studies 61, 397-415.

[29] Peek, J., and E.S. Rosengren (1997), “The International Transmission of Financial

Shocks: The Case of Japan,” American Economic Review 87, 495-505.

[30] Peek, J., and E.S. Rosengren (2000), “Collateral Damage: Effects of the Japanese Bank

Crises on Real Activity in the United States,” American Economic Review 90, 30-45.

[31] Petersen, M.A., and R.G. Rajan (1994), “The Benefits of Lending Relationships: Evi-

dence from Small Business Data,” Journal of Finance 49, 3-37.

[32] Petersen, M.A., and R.G. Rajan (1995), “The Effect of Credit Market Competition on

Lending Relationships,” Quarterly Journal of Economics 110, 407-443.


[33] Prowse, S.D. (1998), “The Economics of the Private Equity Market,” Federal Reserve

Bank of Dallas Review, Third Quarter, 21-34.

[34] Ramey, G., and J. Watson (1997), “Contractual Fragility, Job Destruction, and Business

Cycles,” Quarterly Journal of Economics 112, 873-911.

[35] Ramey, G., and J. Watson (2002), “Contractual Intermediaries,” Journal of Law, Eco-

nomics, and Organization, forthcoming.

[36] Rose, S. (1993), “Comment,” in: Structural Change in Banking, Michael Klausner and

Lawrence J. White (eds), Business One Irwin, Homewood Illinois, 295-299.

[37] Sharpe, S.A. (1995), “Bank Capitalization, Regulation, and the Credit Crunch: A

Critical Review of the Research Findings,” Federal Reserve Board FEDC working paper

95-20.

[38] Slovin, M.B., M.E. Sushka, and J.A. Polonchek (1993), “The Value of Bank Durability:

Borrowers as Bank Stakeholders,” Journal of Finance 48, 247-266.

[39] Townsend, R. (1979), “Optimal Contracts and Competitive Markets with Costly State

Verification,” Journal of Economic Theory 21, 265-293.

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